Superintegrability in a non-conformally-flat space

TL;DR

This paper extends methods for establishing superintegrability from conformally flat spaces to a non-conformally-flat four-dimensional space, revealing the need for curvature corrections and polynomial closure of some symmetry subalgebras.

Contribution

It develops a new approach to prove superintegrability in higher-dimensional non-conformally-flat spaces, expanding the class of known superintegrable systems.

Findings

Superintegrable system on a non-conformally-flat 4D space identified.

Curvature corrections to classical potential are necessary.

Some symmetry subalgebras close polynomially.

Abstract

Superintegrable systems in two- and three-dimensional spaces of constant curvature have been extensively studied. From these, superintegrable systems in conformally flat spaces can be constructed by Staeckel transform. In this paper a method developed to establish the superintegrability of the Tremblay-Turbiner-Winternitz system in two dimensions is extended to higher dimensions and a superintegrable system on a non-conformally-flat four-dimensional space is found. In doing so, curvature corrections to the corresponding classical potential are found to be necessary. It is found that some subalgebras of the symmetry algebra close polynomially.